Estimating angular velocity of rigid body via position Suppose you have current state of a rigid body is $S_t = [\theta_t, \phi_t, \psi_t]$ and your past state is  $S_{t-1} = [\theta_{t-1}, \phi_{t-1}, \psi_{t-1}]$
How would you estimate angular velocity using a linear interpolation model? I read something about using Roderiquez rotation formula as the motion model but I'm wondering how is this different than say direct linear interpolation of the angles and how would the derivative of these angles wrt time relate to the angular velocity vector?
 A: $\def \b {\mathbf}$
Assume that your rotation matrix $~\b S~$ is
\begin{align*}
  &\b S=\b S_z(\psi_t)\,\b S_y(\theta_t)\,\b S_x(\phi_t)
  \end{align*}
from here the angular velocity
\begin{align*}
  &\begin{bmatrix}
     0 & -\omega_z & \omega_y \\
     \omega_z & 0 & -\omega_x \\
     -\omega_y & \omega_x & 0 \\
   \end{bmatrix}
  =\dot{\b{S}}\,\b S^T\quad,\Rightarrow
 &\begin{bmatrix}
   \omega_x \\
   \omega_y \\
   \omega_z \\
 \end{bmatrix}_t
 = \left[ \begin {array}{ccc} 0&-\sin \left( \psi_t \right) &\cos \left(
\psi_t \right) \cos \left( \theta_t \right) \\  0&\cos
 \left( \psi_t \right) &\sin \left( \psi_t \right) \cos \left( \theta_t
 \right) \\  1&0&-\sin \left( \theta_t \right)
\end {array} \right]
\,\begin{bmatrix}
    \dot{\psi} \\
    \dot{\theta} \\
    \dot{\phi} \\
  \end{bmatrix}
  \end{align*}
with
\begin{align*}
     &\begin{bmatrix}
    \dot{\psi} \\
    \dot{\theta} \\
    \dot{\phi} \\
  \end{bmatrix}=\frac{1}{\Delta t}
  \begin{bmatrix}
   \Delta\psi \\
   \Delta\theta \\
    \Delta\phi \\
  \end{bmatrix}
  \end{align*}
